Table Of ContentLecture Notes ni
Mathematics
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8541
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Author
Renate Schaaf
Utah State University, Department of Mathematics and Statistics
Logan, UT 84322-3900, USA
Mathematics Subject Classification (1980): 34B15, 35B32, 34C25
ISBN 3-540-53514-4 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-53514-4 Springer-Verlag NewYork Berlin Heidelberg
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Introduction
For the parameter dependent problem
u"(x) + = o , A>0
(1-1-1)
u(o) = u(1) = o
the following is known from application of general local ([13]) or global ([24])
bifurcation theorems in the case f(0) = 0, f'(0) > 0:
The trivial solution u -- 0 exists for each A. From this trivial solution branch
there is bifurcation of nontrivial solutions in each point (0, A) for which the lin-
eurized problem (1-1-1) has a nontrivial kernel, i.e., for A = iTr/~, i =
1, 2, .... The bifurcating branches, i.e., connected components of nontrivial so-
lutions with bifurcation points in the (A, u)-space, are all unbounded and each
branch consists of solutions (A, u) where u has a number of simple zeroes in
] 0, [ 1 which is characteristic for the branch. In the solution branches bifurcating
from 0 all u are bounded by the first positive and first negative zero of'f.
In applications more information about the shape of solution branches is needed.
It is easy to see that all branches are in fact curves which are at least as smooth
as f is (see below). It is then of interest whether these curves have turns with
respect to the A-direction or not, and if so, how many turning points there are
and where they are located. Let, e.g., the branch of positive and negative solutions
to (1-1-1) look like this:
%.
Figure I.l.1
IV Introduction
Then (I-1-1) is the stationary equation of
(I-1-2)
u(t, )O = u(t, 1) = 0
and the directions of the solution branches determine the stability of the stationary
states as indicated.
In combustion problems there often occurs an equation of the form (I-1-1) de-
scribing intermediate steady states of the temperature distribution u, A then
measures the amount of unburnt substance. In this context turning points of a
branch correspond to ignition and extinction points of the process, and it is of
e.g.[151).
importance whether or not those exist (see
In some cases one can skilfutly choose sub- and supersolutions of (I-1-1) and then
get the result that there exist at least two stable solutions for a certain A -interval.
Then using degree theory there has to be at least one more unstable solution for
any A in this interval. This way it is shown that turning points of the branch have
to exist, but one only gets estimates from below for both the number of solutions
and the number of turns of the branch. It is not possible to give upper estimates
of these numbers without using strong analytical tools.
[31])
We use for this purpose the so called time map T (see of the nonlinearity
I:
First of all by a scaling of x , x = At we can write (I-1-1) in the form
¢'(t) +/(.(t)) = 0
(1-1-3)
~(o) u(:~)
= = o
thus having the parameter in the boundary condition.
If u(t) = U(t,p) is the solution of the initial value problem
u" + f(u) = 0
(I-1-4)
u(0) = 0, u'(0) = p # 0
then we define T(p) = TI(p) to be the first positive t for which U(t,p) is zero
again, if this one exists'for the given p :
Introduction V
~ h ~ E
u
t%.
Figure 1.1.2
T2(p) is the second zero of u, T3(p) the third (existence provided) etc..
Then the solution branches of (I-1-3) are given by the graphs of the Ti :
(A, u) solves (I-1-3) if and only if u = U(., p), Ti(p) is defined for some i = 1, 2,...
and
(I-1-5) ~ = Ti(p).
Stable regions of positive or negative solution branches are u'(0) = p -regions with
pT'(p) > 0, unstable ones correspond to pT'(p) < 0, turning points of branches
correspond to T'(po) = 0, T"(po) ~ 7 O.
Time maps and period maps (see below) have been studied in several papers, see
[3], [5], [8], [9], [22], [25], [26], [31], [33], [35], [36], a list of references which is by
no means complete.
By the implicit function theorem Ti is always at least as regular as f is since
U(Ti(p),p) = o OV(T~(p),p) -- Ipl > O.
The last identity follows since (I-1-4) has a first integral
(I-1-6) (u') 2 5- F(u) --= const = -~p
VI Introduction
for u = U(.,p) and F being the integral of f with F(0) --- 0.
Note that with T and Ti we get all solutions of (I-1-3), not only the branches
bifurcating from 0. Also it is not necessary that we have a trivial solution set via
f(0) = 0. (There are general results for global branches in these cases too, we did
not mention them.)
From (1-1-6) one can derive a formula for T which is a singular integral (see (1-1-
3)). For studying derivatives of T this one is not very useful. Therefore in section
1.1 we derive a time map formula for branches bifurcating from zero which does
not contain a singularity and can be easily differentiated. The idea behind this is
to use a transformation in u which maps the orbits of (I-1-4) in the phase plane
into circles. For solutions in bifurcating branches this can be done since f does
not change sign more than once along the range of such solutions.
Chapter 1 consists entirely "of applications of this time map formula. Section 1.2
recalls results about bifurcation points and bifurcation directions. In 1.3 we use
the time map formula to reprove results by Chafee, Infante ([5]) and Opial ([22])
giving conditions for branches which have only a single turn at the bifurcation
point itself.
Paragraph 1.4 goes a step further and gives conditions on f under which T[' does
not change sign. This results in branches which still have at most one turning
point, but this one can occur away from the bifurcation point (see figure 1.1.2).
This allows for a change of stability along branches, but stability can change at
most twice.
We have tried to find a condition on f for Ti t # 0 which can be verified in
examples. For this we introduce the notion of an A-B-function which is a function
f with
(I-1-7) f,f,,, - 5 (f'')~ < 0 in regions where f' > 0.
ff" - 3(f') 2 < 0 in regions where f' < 0.
It is shown that T[ > ~ 0 for all bifurcating branches of A-B-functions. This class
contains all polynomials without complex zeroes as is shown in 1.5. With this we
have generalized the result of Smoller and Wasserman in [31] which says that the
Ti have at most one critical point which then is a minimum in the case that f is
a cubic f(u) = -u(u - a)(u -/3) with ~( < 0 </3. 1.5 contains more methods
for getting hold of A-B-functions, using, e.g., the Schwarzian of f. See also the
examples discussed near the end of this introduction.
Introduction VII
Section 1.6 deals with the asymptotic behaviour of T as p approaches a boundary
point of its definition set. Some of the results can be used to prove existence of
dead core solutions to certain reaction diffusion problems (see examples 1.6.2 and
3.2.2).
Chapter 2 contains results on the Neumann problem corresponding to (I-I-3)
u" + f(u) = 0
(I-1-8) u'(0) = u'(A) = 0,
and related results on period maps of Hamiltonian systems
(I-1-9) u' = f2(v) v' = -f,(u).
If r is some zero of f then u - r is always a trivial solution of (I-1-8). The
linearization of (I-1-8) about u - r has a nonzero solution if and only if f'(r) > O,
i = irc/~, i = 1, 2, .... These are the bifurcation points of (I-1-8). Solutions
u in the bifurcating branches are bounded by the first zeroes of f below and above
r. For these branches one can show that they have a single turn at the bifurcation
point in the case that f is an A-B-function. The proof uses a Neumann time map
defined analogously to the Dirichlet time map mentioned before.
With about the same method we get monotonicity results for the period map of
(I-1-9), which is the map assigning to E the least period H(E) of a periodic
solution of (I-1-9) with energy E - Fl(u) + F2(v). Such results can be used
to prove bifurcation of subharmonic solutions if (I-1-9) is perturbed by a small
nonautonomous t-periodic term (see [7]). Another application of such results
is the existence and uniqueness of period-4-solutions to the time delay equation
u'(t) + f(u(t - 1)) = 0 with antisymmetric f (see [18]).
In chapter 3 we return to the Dirichlet problem and give results for problem (i-1-3)
where we assume less for f and also consider non bifurcating branches:
In 3.1 we no longer assume f(0) = 0, f'(0) > 0 but just consider some f defined
on ] 0, a [ which is positive there. This way we can handle the branch of (I-1-3)
which consists of solutions (A,u) with u positive and u" not changing sign. For
these we get a time map formula similar to the one in 1.1.
The regularity of f required for the existence of T turns out to be a fairly
weak one. Therefore problems with singularities at u = 0 can be dealt with.
Multiple zeroes of f at 0 can also occur. In this case there is "bifurcation" from
(A = c~,u = 0). If f is an A-B-function then the branch is shown to be again
U-shaped.
VIII Introduction
The case f(0) > 0 shows more difficulties than f(0) = 0: For f(u) = -(u -
o0(u-/9)(u-7) with ~o < 9/ < 0 < 7 it is shown in [31] that the number of
critical points of T is at most two, and exactly two if 9j is close to 0 .So far we
are not able to generalize this to A-B-functions f, though it seems like this result
is true. The only thing we could do is to give a criterion on f such that T has
at most one critical point. In the case f(0) > 0 it is not possible to show this
via a sign of T" or a related expression. Instead we use variation diminishing
properties of integral operators proved in [19], results are to be found in 3.3.
In 3.4 we discuss positive solution branches in which u" changes sign, i.e. u has
at least one zero of f in its range. Under certain assumptions on f we can again
show that branches are U-shaped.
The last chapter discusses some general features of Dirichlet time maps. In 4.1 we
give proofs for the relationship between the sign of pT'(p) and stability. Also there
is a strong connection between T(p) and the energy level of the corresponding
solution of (I-1-3) if we define the energy level via the Liapounov functional
0/
F(u(t)) dr.
: (u'(t)) 2 -
It turns out that E is decreasing along parts of solution branches with pT'(p) > 0
and increasing if pT'(p) < 0. As another result if there are multiple solutions for
a fixed I an unstable solution always has higher energy than the two "nearest"
stable solutions.
In section 4.2 we again only consider nonlinearities f defined on some interval
] 0, a [ which are positive there. We pose the inverse problem if it is possible to
decide for a given curve p ~ T(p) whether or not it is the time map of some f
and if it is possible to calculate f from T, i.e., to get the problem back from its
solution branch. The set of all time maps of such f is characterized by an integral
condition and the operator f ~ T is shown to be invertible.
Applications include proofs of generic properties of such time maps analogous to
[3], [33], as well as estimates for ~-regions of existence and nonexistence for (I-
1-3), and can be found in 4.3 among some games about possible and impossible
time maps.In the appendix, finally, we briefly discuss the method to obtain time
map plots via the computer.
In order to give a flavour of the possible applications results in this monograph
can have let us discuss two specific examples. Please note that the argumentation
used to interprete results is not rigorous up to the detail.
Introduction IX
a is It good illustration to view (I-l-l) as the stationary equation for a population
of size u diffusing on the interval ,0[ ]I which has a hostile environment thus
forcing u(t,O) = u(t, 1) = 0. 1/~ 2 can then be regarded to be the diffusion
coefficient of the population, wheras f+(u) = f(u)/u models tile reproduction
rate consisting of the birth rate minus the death rate.
Let us first consider the example
f+(u) = e -(u-~)2 , a >_ O.
This means that the reproduction rate is maximal for a certain optimal population
size a and declines to 0 as u becomes large but the death rate never exceeds
the birth rate. This means for the model without diffusion
ut = uf+(u)
that any population with a positive intitial size grows unlimited. If we now add
diffusion
1 +
fu+**u~
(I-i-10)
ut = (u)
t, o) = = o
then the behaviour depends on the size of A and is in principal governed by the
bifurcation diagram for the stationary equation (I-1-1).
All stationary nontrivial branches are given by A = Ti(p). In this specific context
only the positive solution branch = A T(p), p > 0, matters since the system is not
going to get near any sign changing steady state if it starts out with u(0, x) > 0
(maximum principle). Section 1.2 gives the bifurcation points (Ai, u -- 0) as
hi = Ti(0) -
and the bifurcation directions as
{0, /even
T'(0) = 2 f" , i odd
~ (f,)2 (0) = - eca "~'
So the positive branch starts out unstable if a > 0.
For the "other end" of the branches we use section 1.6: First of all the definition
set of the Ti is (see section 1.1) D(T1) = ]b-,b+[ with b- = -X/2F(-oo),
+ b = ~ and D(Ti)=]b-,-b-[ for i>2.
X Introduction
&
From proposition 1.6.1 (iv) it follows that limp,b+ T(p)= with
"
c+ = limu+, F{'(ol) -- limu+, f+(u) = O and T(p)= &= with
C- = limu-+-, f+(u) = 0. SO
.
lim T(p) = +CO
p-b+,b-
Formulas (1-6-17) and (1-6-18) then imply for all the other branches that
>
limp+b- ,-b- Ti(p)= +co , i 2.
Let us put together what we have found out by this for the positive branch X =
T(p), p > 0 : It starts out unstable but has to recover stability later since T(p)+
+CO as p + b+ . If we let A* = minplo T(p)> 0 then no nontrivial steady state
exists for X < A* and the zero solution is a global attractor for the system ([14],
[17]).T he population dies out from too fast diffusion and the hostile environment.
For X > XI = Tea'/' the zero solution looses its stability, but the population
could never grow beyond limits: From the Liapounov functional and the fact that
f+(+m) = 0 it follows that any solution of (1-1-10) is a priori bounded. So
diffusion and the zero boundary conditions prevent unlimited growth.
Results in [17] can be used to obtain further information on the dynamical be-
haviour of our system for X > A* if we know that all steady states are hyperb.olic
(this follows from T;'(p)# 0 if X = Ti(p)) and if we know more about the maxi-
mal invariant set A which consists of the unstable manifolds of all steady states
present for our A. For this example this is possible by applying sections 1.3, 1.4,
1.5 and the results about connecting orbits obtained by Brunovskjr and Fiedler in
-
141
Let us fist consider the case a = 0. Then u - & F = uf+'(u) < 0 and we
can use theorem 1.3.2 to obtain that pT:(p) > 0, which completely proves the
following bifurcation diagram:
